Quantum Equivalent Magnetic Fields that Are Not Classically Equivalent

Abstract

We construct pairs of compact K\"ahler-Einstein manifolds $(M_i,g_i,\omega_i)$ ($i=1,2)$ of complex dimension $n$ with the following properties: The canonical line bundle $L_i=\bigwedge^n T^*M_i$ has Chern class $[\omega_i/2\pi]$, and for each integer $k$ the tensor powers $L_1^{\otimes k}$ and $L_2^{\otimes k}$ are isospectral for the bundle Laplacian associated with the canonical connection, while $M_1$ and $M_2$ -- and hence $T^*M_1$ and $T^*M_2$ -- are not homeomorphic. In the context of geometric quantization, we interpret these examples as magnetic fields which are quantum equivalent but not classically equivalent. Moreover, we construct many examples of line bundles $L$, pairs of potentials $Q_1$, $Q_2$ on the base manifold, and pairs of connections $\nabla_1$, $\nabla_2$ on $L$ such that for each integer $k$ the associated Schr\"odinger operators on $L^{\otimes k}$ are isospectral.